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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

On the difference between perfect powers

Jan Turk (1986)

Acta Arithmetica

On the Diophantine equation 1 + x + y = z, with xyz = 2r3s7t.

Leo J. Alex, Lorraine L. Foster (1995)

Forum mathematicum

On the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$

Ruizhou Tong (2021)

Czechoslovak Mathematical Journal

Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2 ∤ x$ , $p \equiv \pm 3 (mod 8),$ the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$ has no positive integer solution except when $p = 3$ or $p$ is of the form $p = 2 a_{0}^{2} + 1$ , where $a_{0} > 1$ is an odd positive integer. (2) if $2 ∤ x$ , $2 ∣ y$ , $y \neq 2, 4,$ then the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$ has no positive integer solution.

On the diophantine equation $a x ² + b^{m} = 4 y ⁿ$

S. Akhtar Arif, Amal S. Al-Ali (2002)

Acta Arithmetica

On the diophantine equation $D ₁ x ² + D ₂ = 2^{n + 2}$

Maohua Le (1993)

Acta Arithmetica

On the Diophantine equation $D_{1} x^{2} + D_{2}^{m} = 4 y^{n}$ .

Mignotte, M. (1997)

Portugaliae Mathematica

On the diophantine equation D₁x⁴ -D₂y² = 1

Maohua Le (1996)

Acta Arithmetica

On the diophantine equation $f (a^{m}, y) = b^{n}$

Pietro Corvaja, Umberto Zannier (2000)

Acta Arithmetica

On the diophantine equation $(\binom{n}{k}) = x^{l}$

K. Győry (1997)

Acta Arithmetica

P. 294, line 14: For “Satz 8” read “Satz 7”, and for “equation (10)” read “equation (13)”.

On the diophantine equation $n (n + 1) . . . (n + k - 1) = b x^{l}$

K. Győry (1998)

Acta Arithmetica

On the Diophantine equation $\frac{q^{n} - 1}{q - 1} = y$

Amir Khosravi, Behrooz Khosravi (2003)

Commentationes Mathematicae Universitatis Carolinae

There exist many results about the Diophantine equation $(q^{n} - 1) / (q - 1) = y^{m}$ , where $m \geq 2$ and $n \geq 3$ . In this paper, we suppose that $m = 1$ , $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y - 1$ is less than or equal to $3$ . Then we solve completely the Diophantine equation $(q^{n} - 1) / (q - 1) = y$ for infinitely many values of $y$ . This result finds frequent applications in the theory of finite groups.

On the diophantine equation w+x+y = z, with wxyz = 2r 3s 5t.

L. J. Alex, L. L. Foster (1995)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we complete the solution to the equation w+x+y = z, where w, x, y, and z are positive integers and wxyz has the form 2r 3s 5t, with r, s, and t non negative integers. Here we consider the case 1 < w ≤ x ≤ y, the remaining case having been dealt with in our paper: On the Diophantine equation 1+ X + Y = Z, Rocky Mountain J. of Math. This work extends earlier work of the authors in the field of exponential Diophantine equations.

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